Problem 6 Let X₁, X2, X20 be a random sample of size n = 20 from an No(u, Σ) population. Specify each of the following completely. (a) The distribution of (X₁-μ)'E-¹(X₁-μ) (b) The distribution of X and √n(X-μ) (c) The distribution of (n-1)S (d) The distribution of n(X-μ)'E-¹(X-μ)
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- Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data are at or below for respiratory rates in breath per minute during the first 3 years of infancy are given by y=101.82411-0.0125995x+0.00013401x2 for awake infants and y=101.72858-0.0139928x+0.00017646x2 for sleeping infants, where x is the age in months. Source: Pediatrics. a. What is the domain for each function? b. For each respiratory rate, is the rate decreasing or increasing over the first 3 years of life? Hint: Is the graph of the quadratic in the exponent opening upward or downward? Where is the vertex? c. Verify your answer to part b using a graphing calculator. d. For a 1- year-old infant in the 95 th percentile, how much higher is the walking respiratory rate then the sleeping respiratory rate? e. f.8. Given the Beta Distribution where p(x) = Calculate the mean and variance 1 B(a, b) x-¹(1-x)b-1 +1 B(a,b) = ¹ x-¹(1 x)b-¹dx = = - r(a) (b) I(a + b)4-52. If f(x) =e ,- co20. A random variable X is said to have a beta distribution and is denoted as X Beta(a, 3) if X has the pdf T(a+B) „a-1(1 – x)²-1, for 0 < I < 1; f(r) = < r(a)r(B) otherwise. Find the mean and variance of X.Q.1 The probability mass function for a discrete random variable X is defined as ((1+0)" (^) 0x; x = 0, 1, 2, 3, ..., n fx(x) = {(1 + 0; e. w. where > 0. Show that it is probability mass function. Find its mean and variance.Prob. 3 Let X be a random variable with cumulative distribution function (cdf) given by (1-e-x², x ≥ 0 ={1,- x<0 Find the probability that the random variable X falls within one standard deviation of its mean. Fx (x) =Suppose you have a joint distribution of x and y where • x has population mean ug and population variance o? • y has population mean ly and population variance o, and the population covariance between them is Ery. The population Pearson correlation is then given by Ery OxJy We collect n pairs of data, (X;, Yi), i = 1,.. , n. Each pair is an independent draw from this distribution. (a) Suppose we estimate our population covariance with n 1 (x; – Ha)(yi – µy). i=1 Is this an unbiased estimator of the population covariance? Show why or why not. (b) Suppose we estimate our correlation with (xi – Ha)(Yi – Hy) - n i=1 Ox0y Is this an unbiased estimator of the population correlation? Show why or why not. =WIF. Suppose that a random variable X has normal distribution with mean µ = 2 and variance σ 2 = 9, that is, X ∼ N(2, 9). (30) E[(X + 2)^2 ] is (a) 20 (b) 25 (c) 15 (d) 30 (31) The variance of X/2 + 3 is (a) 9/4 (b) 3/8 (c) 9 (d) 9/2Suppose you take independent random samples from populations with means µi and µ2 and standard deviations ơ1 and o2. Furthermore, assume either that (i) both populations have normal distributions, or (ii) the sample sizes (ni and n2) are large. If X1 and X2 are the random sample means, then (81-X2)-(41-42) how does the quantity behave? Give the n1 n2 name of the distribution and any parameters needed to describe it.5. The variance of a random variable X is defined as the expected value of the squared deviation from its mean 4 = EX.ie. Var(X) = E[(X-EX])²]. Show that Var(X) = EX ELX2. (a) Let random variables X1 and X2 be independent and have a common distribution of Exponential with mean 1. Find the pdf of Z = X1/X2. (b) What is the name of the distribution of Z?45. (a) Karl Pearson's coefficient of skewness for a distribution is - 0.4, its mean is 50 and coefficient of variation is 40%. Obtain standard deviation, median and mode of the given distribution. (b) For a distribution, mean is 10, standard deviation is 4, B1 = 1 and B. = 4. Obtain the first four %3D moments about zero.